Subgroups — #20276745

Order 6784.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810805	C₂	1	3	yes	maximal
2	#16810422	C₂	1	3	yes	maximal, complemented
3	Q₈²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#13133265	C₂²	1	1	yes	characteristic
5	#13133248	—	2	3	no	—
6	#13133145	—	2	3	no	—
7	#13133136	C₂²	1	3	yes	—
8	C₄⋊Q₈×C₅₃	—	2	3	no	—
9	C₄⋊Q₈×C₅₃	C₂²	1	3	yes	—
10	C₄×Q₈×C₅₃	—	2	3	no	—
11	C₂×Q₈×C₅₃	—	2	1	no	—
12	C₂×Q₈×C₅₃	—	2	1	no	—
13	C₂×Q₈×C₅₃	D₄	1	3	yes	—
14	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
15	#11719949	—	4	1	no	—
16	Q₈×D₅₃	—	4	1	no	—
17	#11719945	—	2	3	no	—
18	C₄⋊C₄×C₅₃	—	2	3	no	—
19	C₄⋊C₄×C₅₃	—	2	6	no	—
20	C₄²×C₅₃	—	2	3	no	—
21	C₄²×C₅₃	D₄	1	3	yes	—
22	#11719925	—	4	3	no	—
23	#11719924	—	4	3	no	—
24	#11719917	—	2	3	no	—
25	Q₈×C₅₃	—	2	4	no	—
26	Q₈×C₅₃	—	4	1	no	—
27	Q₈×C₅₃	—	4	3	no	—
28	Q₈×C₅₃	—	8	1	no	—
29	C₂×C₄×C₅₃	—	2	3	no	—
30	C₂×C₄×C₅₃	—	2	3	no	—
31	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
32	C₂²⋊D₅₃	—	4	1	no	—
33	D₂₁₂	—	2	3	no	—
34	C₄×D₅₃	—	4	3	no	—
35	Dic₁₀₆	—	2	3	no	—
36	#88322	—	4	3	no	—
37	C₂²×C₅₃	#120	1	1	yes	characteristic
38	C₂×D₅₃	—	4	1	no	—
39	C₄×C₅₃	—	2	6	no	—
40	C₄×C₅₃	—	2	3	no	—
41	C₄×C₅₃	—	4	3	no	—
42	Dic₅₃	—	4	1	no	—
43	#2205	—	53	1	no	maximal, Sylow
44	C₂×C₅₃	—	2	1	no	—
45	C₂×C₅₃	B(4,𝔽₂)	1	1	yes	characteristic
46	D₅₃	—	8	1	no	—
47	Q₈²	D₅₃	1	1	yes	characteristic, complemented
48	#456	—	53	3	no	—
49	#356	—	53	3	no	—
50	C₅₃	#2205	1	1	yes	characteristic, complemented, Sylow
51	2₋¹⁺⁴	—	53	1	no	—
52	#137	—	106	3	no	—
53	C₄⋊Q₈	—	2	3	no	—
54	C₄⋊Q₈	C₂×D₅₃	1	3	yes	—
55	C₄×Q₈	—	2	3	no	—
56	#104	—	106	3	no	—
57	#101	—	53	3	no	—
58	#41	—	106	3	no	—
59	#41	—	212	1	no	—
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	212	1	no	—
63	C₂×Q₈	C₂²⋊D₅₃	1	3	yes	—
64	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
65	Q₁₆	—	212	3	no	—
66	SD₁₆	—	212	3	no	—
67	M₁₆	—	106	3	no	—
68	C₄⋊C₄	—	2	3	no	—
69	C₄⋊C₄	—	2	6	no	—
70	C₄²	—	2	3	no	—
71	C₄²	C₂²⋊D₅₃	1	3	yes	—
72	Q₈	—	2	4	no	—
73	Q₈	—	4	1	no	—
74	Q₈	—	4	3	no	—
75	Q₈	—	8	1	no	—
76	Q₈	—	106	3	no	—
77	D₄	—	106	3	no	—
78	D₄	—	212	1	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	212	3	no	—
82	C₂×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
83	C₈	—	212	3	no	—
84	C₂²	—	212	1	no	—
85	C₂²	#13133186	1	1	yes	characteristic
86	C₄	—	2	6	no	—
87	C₄	—	2	3	no	—
88	C₄	—	4	3	no	—
89	C₄	—	212	1	no	—
90	C₂	—	2	1	no	—
91	C₂	—	424	1	no	—
92	C₂	#16810810	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810810
commutator	C₂×Q₈×C₅₃	C₂³
Frattini	C₂×Q₈	C₂²×D₅₃
Fitting	Q₈²×C₅₃	C₂
socle	C₂×C₅₃	B(4,𝔽₂)
hypercenter	Q₈²	D₅₃
hypocenter	C₅₃	#2205
LCS	G ▷ C₂×Q₈×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ Q₈²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #16810422	C₂	semidirect
3. Q₈²×C₅₃	C₂	semidirect
47. Q₈²	D₅₃	semidirect
50. C₅₃	#2205	semidirect